Questions: Key Features of Polynomial Functions: Mastery Test Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s). The degree of the function f(x)=-(x+1)^2(2x-3)(x+2)^2 is , and its y-intercept is ( ).

Solution Steps

To solve this problem, we need to determine two things: the degree of the polynomial function and its y-intercept.

1. Degree of the Polynomial: The degree of a polynomial is the highest power of \( x \) in the expanded form of the polynomial. For the given function \( f(x)=-(x+1)^{2}(2x-3)(x+2)^{2} \), we can find the degree by summing the exponents of \( x \) in each term.
2. Y-Intercept: The y-intercept is the value of the function when \( x = 0 \). We substitute \( x = 0 \) into the function and simplify to find the y-intercept.

The function given is \( f(x) = -(x + 1)^{2}(2x - 3)(x + 2)^{2} \). To find the degree, we analyze the exponents of each factor:

• The term \( (x + 1)^{2} \) contributes a degree of \( 2 \).
• The term \( (2x - 3) \) contributes a degree of \( 1 \).
• The term \( (x + 2)^{2} \) contributes a degree of \( 2 \).

Adding these together, we have: \[ \text{Degree} = 2 + 1 + 2 = 5 \]

To find the y-intercept, we evaluate the function at \( x = 0 \): \[ f(0) = -((0 + 1)^{2})(2(0) - 3)((0 + 2)^{2}) = -(1^{2})(-3)(2^{2}) = -1 \cdot -3 \cdot 4 = 12 \]

Final Answer